Moments of the ground state density for the $d$-dimensional Fermi gas in an harmonic trap

We consider properties of the ground state density for the $d$-dimensional Fermi gas in an harmonic trap. Previous work has shown that the $d$-dimensional Fourier transform has a very simple functional form. It is shown that this fact can be used to deduce that the density itself satisfies a third order linear differential equation, previously known in the literature but from other considerations. It is shown too how this implies a closed form expression for the $2k$-th non-negative integer moments of the density, and a second order recurrence. Both can be extended to general Re$\, k > -d/2$. The moments, and the smoothed density, permit expansions in $1/\tilde{M}^2$, where $\tilde{M} = M + (d+1)/2$, with $M$ denoting the shell label. The moment expansion substituted in the second order recurrence gives a generalisation of the Harer--Zagier recurrence, satisfied by the coefficients of the $1/N^2$ expansion of the moments of the spectral density for the Gaussian unitary ensemble in random matrix theory.